Vector Analysis for Minimum Temperature on a Curve: Solve with Grad(p) Method

In summary, the conversation discusses finding the minimum value of the temperature function, p(x,y,z), at a point on a given curve using vector methods. The suggested methods include using the method of Lagrange multipliers and substituting the curve's coordinates into the temperature function. The conversation also mentions the importance of understanding the relationship between the curve and the temperature function in order to accurately determine the minimum value. In more complex situations, numerical methods may be necessary.
  • #1
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Homework Statement



By vector methods, find the point on the curve x = t, y = t^2, z = 2 at which the temperature p(x,y,z) = x^2 - 6x + y^2 takes its minimum value

Homework Equations





The Attempt at a Solution


As far as I got was finding grad(p). From there I'm not sure where to go -- If I take the magnitude of grad(p) i can find the maximum at any point in space, but how to find the minimum I have no clue. I'm also just having trouble visualizing what exactly is being asked and how a curve and the surface relate to a physical quantity like the temperature. If anyone could help me out that would be much appreciated. Thanks a lot
 
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  • #2
I believe this calls for the method of Lagrange multipliers?
 
  • #3
If r(t) is your curve, the 'vector' way is to find points where r'(t).grad(p)=0. Do you see why? grad(p) points in direction of steepest change for p, the direction normal to that is the derivative 0 direction. The direct way to do it is just to substitute the r(t) coordinate values into p, and then find the minimum as a function of t. Try is both ways. Algebraically they are identical.
 
  • #4
doing that i get the following:

grad(p) = <2x - 6, 2y, 0>
dr/dt = <1, 2t, 0>

grad(p) . dr/dt = 2x - 6 + 4ty = 0.

From this i see that the point where this is true is (1,1,2) with t = 1, which is the right answer. However, I am still somewhat confused because I have no clue how I would pick a correct value for the parameter given a more complicated situation since I chose it because it was trivial. I feel like i just got lucky in picking the correct answer. Is there a more "algorithmic" way to do this? Thank you for all the help!
 
  • #5
In a more complicated situation, the situation is more complicated. Then you might have to resort to numerical methods instead of just guessing the answer. That's life. It can be complicated. This is an exercise. It's not complicated.
 

Related to Vector Analysis for Minimum Temperature on a Curve: Solve with Grad(p) Method

1. What is Vector Analysis for Minimum Temperature on a Curve?

Vector analysis for minimum temperature on a curve is a mathematical method used to solve for the minimum temperature of a curved surface or object. It involves using the gradient of the temperature function to find the direction of steepest descent, and then minimizing the temperature along that direction to find the minimum value.

2. What is the Grad(p) Method?

The Grad(p) method is a mathematical technique used in vector analysis for minimum temperature on a curve. It involves taking the gradient of the temperature function and setting it equal to zero to find the minimum temperature point on the curve. This method is based on the fact that the gradient of a function points in the direction of steepest ascent, and the opposite direction of steepest descent.

3. How is Vector Analysis for Minimum Temperature on a Curve useful?

Vector analysis for minimum temperature on a curve is useful in many scientific and engineering fields. It can be used to optimize the design of heat exchangers, predict temperature changes in materials, and analyze heat transfer in various systems. It can also be used to solve practical problems, such as minimizing energy consumption in heating and cooling systems.

4. What are the limitations of Vector Analysis for Minimum Temperature on a Curve?

There are some limitations to using vector analysis for minimum temperature on a curve. One limitation is that it assumes a smooth, continuous temperature function, which may not always be the case in real-world scenarios. Additionally, it requires knowledge of the temperature function and its gradient, which may not always be readily available. It also assumes that the minimum temperature lies on the curve, which may not always be true.

5. Can Vector Analysis for Minimum Temperature on a Curve be applied to 3D surfaces?

Yes, vector analysis for minimum temperature on a curve can be applied to 3D surfaces. In fact, it is often used to optimize the design of 3D objects such as heat sinks and heat exchangers. However, the calculations become more complex in 3D, as the gradient vector now has three components instead of two. Specialized software or programming may be necessary to solve for the minimum temperature in these cases.

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